Not sure what you mean by a "bunch of syntax errors". I get just one syntax error, which describes exactly what the problem is (N is not an infix operator) and points to the place in the input where the error occurred. I don't know why you'd expect a syntax error from nesting an integral inside a limit.
I only wish more of our error messages were so precise!
(%i1) limit(1 / n^2 * integrate(bfpsi(-2,x+1,16) * (sin((2n - 1)*x) / sin(x))^3, x, 0, %pi / 2), n, inf);
incorrect syntax: N is not an infix operator
,x+1,16) * (sin((2n